Next Article in Journal
Decomposition and Arrow-Like Aggregation of Fuzzy Preferences
Next Article in Special Issue
The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces
Previous Article in Journal
Some Fractional Dynamic Inequalities of Hardy’s Type via Conformable Calculus
Previous Article in Special Issue
Existence of a Unique Fixed Point for Nonlinear Contractive Mappings

Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

# A Strong Convergence Theorem under a New Shrinking Projection Method for Finite Families of Nonlinear Mappings in a Hilbert Space

by
Wataru Takahashi
1,2,3
1
Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan
2
Keio Research and Education Center for Natural Sciences, Keio University, Kouhoku-ku, Yokohama 223-8521, Japan
3
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8552, Japan
Mathematics 2020, 8(3), 435; https://doi.org/10.3390/math8030435
Submission received: 2 January 2020 / Accepted: 8 March 2020 / Published: 17 March 2020

## Abstract

:
In this paper, using a new shrinking projection method, we deal with the strong convergence for finding a common point of the sets of zero points of a maximal monotone mapping, common fixed points of a finite family of demimetric mappings and common zero points of a finite family of inverse strongly monotone mappings in a Hilbert space. Using this result, we get well-known and new strong convergence theorems in a Hilbert space.
MSC:
47H05; 47H10

## 1. Introduction

Let H be a real Hilbert space and let C be a nonempty, closed and convex subset of H. Let $T : C → H$ be a mapping. Then we denote by $F ( T )$ the set of fixed points of T. For a real number t with $0 ≤ t < 1$, a mapping $U : C → H$ is said to be a t-strict pseudo-contraction [1] if
$∥ U x − U y ∥ 2 ≤ ∥ x − y ∥ 2 + t ∥ x − U x − ( y − U y ) ∥ 2$
for all $x , y ∈ C$. In particular, if $t = 0$, then U is nonexpansive, i.e.,
$∥ U x − U y ∥ ≤ ∥ x − y ∥ , ∀ x , y ∈ C .$
If U is a t-strict pseudo-contraction and $F ( U ) ≠ ∅$, then we get that, for $x ∈ C$ and $p ∈ F ( U )$,
$∥ U x − p ∥ 2 ≤ ∥ x − p ∥ 2 + t ∥ x − U x ∥ 2 .$
From this inequality, we get that
$∥ U x − x ∥ 2 + ∥ x − p ∥ 2 + 2 〈 U x − x , x − p 〉 ≤ ∥ x − p ∥ 2 + t ∥ x − U x ∥ 2 .$
Then we get that
$2 〈 x − U x , x − p 〉 ≥ ( 1 − t ) ∥ x − U x ∥ 2 .$
A mapping $U : C → H$ is said to be generalized hybrid [2] if there exist real numbers $α , β$ such that
$α ∥ U x − U y ∥ 2 + ( 1 − α ) ∥ x − U y ∥ 2 ≤ β ∥ U x − y ∥ 2 + ( 1 − β ) ∥ x − y ∥ 2$
for all $x , y ∈ C$. Such a mapping U is said to be ($α$, $β$)-generalized hybrid. The class of generalized hybrid mappings covers several well-known mappings. A (1,0)-generalized hybrid mapping is nonexpansive. For $α = 2$ and $β = 1$, it is nonspreading [3,4], i.e.,
$2 ∥ U x − U y ∥ 2 ≤ ∥ U x − y ∥ 2 + ∥ U y − x ∥ 2 , ∀ x , y ∈ C .$
For $α = 3 2$ and $β = 1 2$, it is also hybrid [5], i.e.,
$3 ∥ U x − U y ∥ 2 ≤ ∥ x − y ∥ 2 + ∥ U x − y ∥ 2 + ∥ U y − x ∥ 2 , ∀ x , y ∈ C .$
In general, nonspreading mappings and hybrid mappings are not continuous; see [6]. If U is a generalized hybrid and $F ( U ) ≠ ∅$, then we get that, for $x ∈ C$ and $p ∈ F ( U )$,
$α ∥ p − U x ∥ 2 + ( 1 − α ) ∥ p − U x ∥ 2 ≤ β ∥ p − x ∥ 2 + ( 1 − β ) ∥ p − x ∥ 2$
and hence $∥ U x − p ∥ 2 ≤ ∥ x − p ∥ 2 .$ From this, we have that
$2 〈 x − p , x − U x 〉 ≥ ∥ x − U x ∥ 2 .$
We also know that such a mapping exists in a Banach space. Let E be a smooth Banach space and let G be a maximal monotone mapping with $G − 1 0 ≠ ∅$. Then, for the metric resolvent $J λ$ of G for a positive number $λ > 0$, we obtain from [7,8] that, for $x ∈ E$ and $p ∈ G − 1 0 = F ( J λ )$,
$〈 J λ x − p , J ( x − J λ x ) 〉 ≥ 0 .$
Then we get
$〈 J λ x − x + x − p , J ( x − J λ x ) 〉 ≥ 0$
and hence
$〈 x − p , J ( x − J λ x ) 〉 ≥ ∥ x − J λ x ∥ 2 ,$
where J is the duality mapping on E. Motivated by (1), (2) and (3), Takahashi [9] introduced a nonlinear mapping in a Banach space as follows: Let C be a nonempty, closed, and convex subset of a smooth Banach E and let $η$ be a real number with $η ∈ ( − ∞ , 1 )$. A mapping $U : C → E$ with $F ( U ) ≠ ∅$ is said to be $η$-demimetric if, for $x ∈ C$ and $p ∈ F ( U )$,
$2 〈 x − p , J ( x − U x ) 〉 ≥ ( 1 − η ) ∥ x − U x ∥ 2 .$
According to this definition, we have that a t-strict pseudo-contraction U with $F ( U ) ≠ ∅$ is t-demimetric, an ($α$, $β$)-generalized hybrid mapping U with $F ( U ) ≠ ∅$ is 0-demimetric and the metric resolvent $J λ$ with $G − 1 0 ≠ ∅$ is $( − 1 )$-demimetric. On the other hand, we know the shrinking projection method which was defined by Takahashi, Takeuchi, and Kubota [10] for finding fixed points of nonexpansive mappings in a Hilbert space. They proved the following strong convergence theorem [10].
Theorem 1
([10]).Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let$U : C → C$be a nonexpansive mapping. Assume that$F ( U ) ≠ ∅$. For$x 1 ∈ C$and$C 1 = C$, let${ x n }$be a sequence defined by
$y n = ( 1 − λ n ) x n + λ n U x n , C n + 1 = { z ∈ C n : ∥ y n − z ∥ ≤ ∥ x n − z ∥ } , x n + 1 = P C n + 1 x 1 , n = 1 , 2 , … . ,$
where a real number a and${ λ n } ⊂ ( 0 , ∞ )$satisfy the following inequalities:
$0 < a ≤ λ n ≤ 1 , n = 1 , 2 , … .$
Then the sequence${ x n }$converges strongly to$u ∈ F ( U )$, where$u = P F ( U ) x 1$and$P F ( U )$is the metric projection of H onto$F ( U )$.
In this paper, using a new shrinking projection method, we prove a strong convergence theorem for finding a common point of the sets of zero points of a maximal monotone mapping, common fixed points for a finite family of demimetric mappings and common zero points of a finite family of inverse strongly monotone mappings in a Hilbert space. Using this result, we obtain well-known and new strong convergence theorems in a Hilbert space. In particular, using the shrinking projection method, we prove a strong convergence theorem for a finite family of generalized hybrid mappings with the variational inequalty problem in a Hilbert space.

## 2. Preliminaries

Throughout this paper, let H be a real Hilbert space with inner product $· , ·$ and norm $·$ and let $N$ and $R$ be the sets of positive integers and real numbers, respectively. When ${ x n }$ is a sequence in H, we denote by $x n → x$ the strong convergence of ${ x n }$ to $x ∈ H$ and by $x n ⇀ x$ the weak convergence. We have from [11,12] that, for $x , y ∈ H$ and $α ∈ R$,
$∥ α x + ( 1 − α ) y ∥ 2 = α ∥ x ∥ 2 + ( 1 − α ) ∥ y ∥ 2 − α ( 1 − α ) ∥ x − y ∥ 2 .$
Furthermore, we have that, for $x , y , u , v ∈ H$,
$2 〈 x − y , u − v 〉 = ∥ x − v ∥ 2 + ∥ y − u ∥ 2 − ∥ x − u ∥ 2 − ∥ y − v ∥ 2 .$
Let C be a nonempty, closed and convex subset of H. A mapping $U : C → H$ with $F ( U ) ≠ ∅$ is said to be quasi-nonexpansive if $∥ U x − p ∥ ≤ ∥ x − p ∥$ for all $x ∈ C$ and $p ∈ F ( U )$. If $U : C → H$ is quasi-nonexpansive, then $F ( U )$ is closed and convex; see [12,13]. For a nonempty, closed, and convex subset D of H, the nearest point projection of H onto D is denoted by $P D$, that is,
$x − P D x ≤ x − y , ∀ x ∈ H , y ∈ D .$
A mapping $P D$ is said to be the metric projection of H onto D. The inequality (6) is equivalent to
$x − P D x , y − P D x ≤ 0 , ∀ x ∈ H , y ∈ D .$
We obtain from (7) that $P D$ is firmly nonexpansive, that is,
$P D x − P D y 2 ≤ 〈 P D x − P D y , x − y 〉 , ∀ x , y ∈ H .$
In fact, from (7) we have that, for $x . y ∈ H$,
$x − P D y + P D y − P D x , P D y − P D x ≤ 0$
and hence
$P D x − P D y 2 ≤ 〈 P D x − P D y , x − P D y 〉 = 〈 P D x − P D y , x − y + y − P D y 〉 = 〈 P D x − P D y , x − y 〉 + 〈 P D x − P D y , y − P D y 〉 ≤ 〈 P D x − P D y , x − y 〉 .$
Furthermore, using (7) and (5), we have that
$∥ P D x − y ∥ 2 + ∥ P D x − x ∥ 2 ≤ ∥ x − y ∥ 2 , ∀ x ∈ H , y ∈ D .$
Let C be a nonempty, closed, and convex subset of H. A mapping $A : C → H$ is said to be $α$-inverse strongly monotone if there exists $α > 0$ such that
$〈 x − y , A x − A y 〉 ≥ α ∥ A x − A y ∥ 2 , ∀ x , y ∈ C .$
If A is an $α$-inverse-strongly monotone mapping and $0 < μ ≤ 2 α$, then we obtain from [12] that $I − μ A : C → H$ is nonexpansive, i.e.,
$∥ ( I − μ A ) x − ( I − μ A ) y ∥ ≤ ∥ x − y ∥ , ∀ x , y ∈ C .$
For more results of inverse strongly monotone mappings, see also [12,14,15]. The variational inequalty problem for a nonlinear mapping $A : C → H$ is to find an element $w ∈ C$ such that
$〈 A w , x − w 〉 ≥ 0 , ∀ x ∈ C .$
The set of solutions of (10) is denoted by $V I ( C , A )$. We also have that, for $μ > 0$, $w = P C ( I − μ A ) w$ if and only if $w ∈ V I ( C , A )$. In fact, let $μ > 0$. Then, for $w ∈ C$,
$w = P C ( I − μ A ) w ⟺ 〈 ( I − μ A ) w − w , w − y 〉 ≥ 0 , ∀ y ∈ C ⟺ 〈 − μ A w , w − y 〉 ≥ 0 , ∀ y ∈ C ⟺ 〈 A w , w − y 〉 ≤ 0 , ∀ y ∈ C ⟺ 〈 A w , y − w 〉 ≥ 0 , ∀ y ∈ C ⟺ w ∈ V I ( C , A ) .$
Let G be a multi-valued mapping from H into H. The effective domain of G is denoted by $dom ( G )$, i.e., $dom ( G ) = { x ∈ H : G x ≠ ∅ }$. A multi-valued mapping $G ⊂ H × H$ is called a monotone mapping on H if $x − y , u − v ≥ 0$ for all $x , y ∈ dom ( G )$, $u ∈ G x$, and $v ∈ G y$. A monotone mapping G on H is said to be maximal if its graph is not properly contained in the graph of any other monotone mapping on H. For a maximal monotone mapping G on H, we may define a single-valued mapping $J r = ( I + r G ) − 1 : H → dom ( G )$, which is said to be the resolvent of G for $r > 0$. We denote by $A r = 1 r ( I − J r )$ the Yosida approximation of G for $r > 0$. We get from [8] that
$A r x ∈ G J r x , ∀ x ∈ H , r > 0 .$
For a maximal monotone mapping G on H, let $G − 1 0 = { x ∈ H : 0 ∈ G x } .$ It is known that $G − 1 0 = F ( J r )$ for all $r > 0$ and the resolvent $J r$ is firmly nonexpansive:
$∥ J r x − J r y ∥ 2 ≤ 〈 J r x − J r y , x − y 〉 , ∀ x , y ∈ H .$
Takahashi, Takahashi, and Toyoda [16] proved the following result.
Lemma 1
([16]).Let G be a maximal monotone mapping on a Hilbert space H. For$r > 0$and$x ∈ H$, define the resolvent$J r x$. Then the following inequality holds:
$s − t s 〈 J s x − J t x , J s x − x 〉 ≥ ∥ J s x − J t x ∥ 2$
for all$s , t > 0$and$x ∈ H$.
From Lemma 1, we get that, for $s , t > 0$ and $x ∈ H$,
$∥ J s x − J t x ∥ 2 ≤ | s − t | s ∥ J s x − x ∥ ∥ J s x − J t x ∥$
and hence
$∥ J s x − J t x ∥ ≤ | s − t | s ∥ J s x − J t x ∥ .$
Using the ideas of [17,18], Alsulami and Takahashi [19] proved the following lemma.
Lemma 2
([19]).Let C be a nonempty, closed and convex subset of a Hilbert space H. Let$G ⊂ H × H$be a maximal monotone mapping and let$J λ = ( I + λ G ) − 1$be the resolvent of G for$λ > 0$. Let$κ > 0$and let$U : C → H$be a κ-inverse strongly monotone mapping. Suppose that$G − 1 0 ∩ U − 1 0 ≠ ∅$. Let$λ , r > 0$and$z ∈ C$. Then the following are equivalent:
(i)
$z = J λ ( I − r U ) z$;
(ii)
$0 ∈ U z + G z$;
(iii)
$z ∈ G − 1 0 ∩ U − 1 0$.
When a Banach space E is a Hilbert space, the definition of a demimetric mapping is as follows: Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let $η ∈ ( − ∞ , 1 )$. A mapping $U : C → H$ with $F ( U ) ≠ ∅$ is said to be $η$-demimetric [9] if, for $x ∈ C$ and $q ∈ F ( U )$,
$〈 x − q , x − U x 〉 ≥ 1 − η 2 ∥ x − U x ∥ 2 .$
The following lemma which was essentially proved in [9] is important and crucial in the proof of the main result. For the sake of completeness, we give the proof.
Lemma 3
([9]).Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let η be a real number with$η ∈ ( − ∞ , 1 )$and let U be an η-demimetric mapping of C into H. Then$F ( U )$is closed and convex.
Proof.
Let us show that $F ( U )$ is closed. For a sequence ${ q n }$ such that $q n → q$ and $q n ∈ F ( U )$, we have from the definition of U that
$2 〈 q − q n , q − U q 〉 ≥ ( 1 − η ) ∥ q − U q ∥ 2 .$
From $q n → q$, we have $0 ≥ ( 1 − η ) ∥ q − U q ∥ 2$. From $1 − η > 0$, we have $∥ q − U q ∥ = 0$ and hence $q = U q$. This implies that $F ( U )$ is closed.
Let us prove that $F ( U )$ is convex. Let $p , q ∈ F ( U )$ and set $z = α p + ( 1 − α ) q$, where $α ∈ [ 0 , 1 ]$. Then we have that
$2 〈 z − p , z − U z 〉 ≥ ( 1 − η ) ∥ z − U z ∥ 2 and 2 〈 z − q , z − U z 〉 ≥ ( 1 − η ) ∥ z − U z ∥ 2 .$
From $α ≥ 0$ and $1 − α ≥ 0$, we also have that
$2 〈 α z − α p , z − U z 〉 ≥ α ( 1 − η ) ∥ z − U z ∥ 2$
and $2 〈 ( 1 − α ) z − ( 1 − α ) q , z − U z 〉 ≥ ( 1 − α ) ( 1 − η ) ∥ z − U z ∥ 2 .$ > From these inequalities, we get that
$0 = 2 〈 z − z , z − U z 〉 ≥ ( 1 − η ) ∥ z − U z ∥ 2 .$
From $1 − η > 0$ we get that $∥ z − U z ∥ = 0$ and hence $z = U z$. This means that $F ( U )$ is convex. ☐
Takahashi, Wen, and Yao [20] proved the following lemma which is also used in the proof of the main result.
Lemma 4
([20]).Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let$η ∈ ( − ∞ , 1 )$and let a mapping$T : C → H$with$F ( T ) ≠ ∅$be η-demimetric. Let μ be a real number with$0 < μ ≤ 1 − η$and define$U = ( 1 − μ ) I + μ T$. Then U is a quasi-nonexpansive mapping of C into H.

## 3. Main Result

In this section, using a new shrinking projection method, we obtain a strong convergence theorem for finding a common point of the sets of zero points of a maximal monotone mapping, common fixed points for a finite family of demimetric mappings and common zero points of a finite family of inverse strongly monotone mappings in a Hilbert space. Let C be a nonempty, closed and convex subset of a Hilbert space H. Then a mapping $T : C → H$ is said to be demiclosed if, for a sequence ${ x n }$ in C such that $x n ⇀ w$ and $x n − T x n → 0$, $w = T w$ holds; see [21].
Theorem 2.
Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let${ k 1 , … , k M } ⊂ ( − ∞ , 1 )$and${ μ 1 , … , μ N } ⊂ ( 0 , ∞ )$. Let${ T j } j = 1 M$be a finite family of$k j$-demimetric and demiclosed mappings of C into itself and let${ B i } i = 1 N$be a finite family of$μ i$-inverse strongly monotone mappings of C into H. Let A and G be maximal monotone mappings on H and let$J r = ( I + r A ) − 1$and$Q λ = ( I + λ G ) − 1$be the resolvents of A and G for$r > 0$ and $λ > 0$, respectively. Assume that
$Ω = A − 1 0 ∩ ( ∩ j = 1 M F ( T j ) ) ∩ ( ∩ i = 1 N ( B i + G ) − 1 0 ) ≠ ∅ .$
For$x 1 ∈ C$and$C 1 = C$, let${ x n }$be a sequence defined by
$y n = ∑ j = 1 M ξ j ( ( 1 − λ n ) I + λ n T j ) x n , z n = ∑ i = 1 N σ i Q η n ( I − η n B i ) y n , u n = J r n z n , C n + 1 = { z ∈ C n : ∥ y n − z ∥ ≤ ∥ x n − z ∥ , ∥ z n − z ∥ ≤ ∥ y n − z ∥ a n d 〈 z n − z , z n − u n 〉 ≥ ∥ z n − u n ∥ 2 } , x n + 1 = P C n + 1 x 1 , ∀ n ∈ N ,$
where${ λ n } , { η n } , { r n } ⊂ ( 0 , ∞ )$,${ ξ 1 , … , ξ M } , { σ 1 , … , σ N } ⊂ ( 0 , 1 )$and$a , b , c ∈ R$satisfy the following:
(1)
$0 < a ≤ λ n ≤ min { 1 − k 1 , … , 1 − k M } , ∀ n ∈ N$;
(2)
$0 < b ≤ η n ≤ 2 min { μ 1 , … , μ N } , ∀ n ∈ N$;
(3)
$0 < c ≤ r n , ∀ n ∈ N$;
(4)
$∑ j = 1 M ξ j = 1$and$∑ i = 1 N σ i = 1$.
Then${ x n }$converges strongly to a point$z 0 ∈ Ω$, where$z 0 = P Ω x 1$.
Proof.
Since a mapping $B i$ is $μ i$-inverse strongly monotone for all $i ∈ { 1 , … , N }$ and $0 < b ≤ η n ≤ 2 μ i$, we have that $Q η n ( I − η n B i )$ is nonexpansive and
$F ( Q η n ( I − η n B i ) ) = ( B i + G ) − 1 0$
is closed and convex. Furthermore, we have from Lemma 3 that $F ( T j )$ is closed and convex. We also know that $A − 1 0$ is closed and convex. Then,
$Ω = A − 1 0 ∩ ( ∩ j = 1 M F ( T j ) ) ∩ ( ∩ i = 1 N ( B i + G ) − 1 0 )$
is nonempty, closed, and convex. Therefore, $P Ω$ is well defined.
We have that
$∥ y n − z ∥ ≤ ∥ x n − z ∥ ⟺ ∥ y n − z ∥ 2 ≤ ∥ x n − z ∥ 2 ⟺ ∥ y n ∥ 2 − ∥ x n ∥ 2 − 2 〈 y n − x n , z 〉 ≤ 0 .$
Similarly, we have that
$∥ z n − z ∥ ≤ ∥ y n − z ∥ ⟺ ∥ z n ∥ 2 − ∥ y n ∥ 2 − 2 〈 z n − y n , z 〉 ≤ 0 .$
Thus ${ z ∈ C : ∥ y n − z ∥ ≤ ∥ x n − z ∥ and ∥ z n − z ∥ ≤ ∥ y n − z ∥ }$ is closed and convex. We also have that ${ z ∈ C : 〈 z n − z , z n − u n 〉 ≥ ∥ z n − u n ∥ 2 }$ is closed and convex. Then $C n$ is closed and convex for all $n ∈ N$. Let us show that $Ω ⊂ C n$ for all $n ∈ N$. We have that $Ω ⊂ C 1 = C .$ Assume that $Ω ⊂ C k$ for some $k ∈ N$. From Lemma 4 we have that, for $z ∈ Ω$,
$∥ y k − z ∥ = ∥ ∑ j = 1 M ξ j ( ( 1 − λ k ) I + λ k T j ) x k − z ∥ ≤ ∑ j = 1 M ξ j ∥ ( ( 1 − λ k ) I + λ k T j ) x k − z ∥ ≤ ∑ j = 1 M ξ j ∥ x k − z ∥ = ∥ x k − z ∥ .$
Furthermore, since $Q η k ( I − η k B i )$ is nonexpansive and hence quasi-nonexpansive, we have that, for $z ∈ Ω$,
$∥ z k − z ∥ = ∥ ∑ i = 1 N σ i Q η k ( I − η k B i ) y k − z ∥ ≤ ∑ i = 1 N σ i ∥ Q η k ( I − η k B i ) y k − z ∥ ≤ ∑ i = 1 N σ i ∥ y k − z ∥ = ∥ y k − z ∥ .$
Since $J r k$ is the resolvent of A and $u k = J r k z k$, we also have that
$〈 z k − J r k z k , J r k z k − z 〉 ≥ 0 , ∀ z ∈ Ω .$
From $〈 z k − J r k z k , J r k z k − z k + z k − z 〉 ≥ 0$, we have that
$〈 z k − J r k z k , z k − z 〉 ≥ ∥ z k − J r k z k ∥ 2 .$
This implies that
$〈 z k − u k , z k − z 〉 ≥ ∥ z k − u k ∥ 2 .$
From these, we have that $Ω ⊂ C k + 1$. Therefore, we have by mathematical induction that $Ω ⊂ C n$ for all $n ∈ N$. Thus $x n + 1 = P C n + 1 x 1$ is well defined.
Since $Ω$ is nonempty, closed, and convex, there exists $z 0 ∈ Ω$ such that $z 0 = P Ω x 1$. By $x n + 1 = P C n + 1 x 1$, we get that
$∥ x 1 − x n + 1 ∥ ≤ ∥ x 1 − z ∥$
for all $z ∈ C n + 1$. From $z 0 ∈ Ω ⊂ C n + 1$ we obtain that
$∥ x 1 − x n + 1 ∥ ≤ ∥ x 1 − z 0 ∥ .$
This implies that ${ x n }$ is bounded. Since $x n = P C n x 1$ and $x n + 1 ∈ C n + 1 ⊂ C n$, we get that
$∥ x 1 − x n ∥ ≤ ∥ x 1 − x n + 1 ∥ .$
Thus ${ ∥ x 1 − x n ∥ }$ is bounded and nondecreasing. Then the limit of ${ ∥ x 1 − x n ∥ }$ exists. Put $lim n → ∞ ∥ x n − x 1 ∥ = c$. For any $m , n ∈ N$ with $m ≥ n$, we have $C m ⊂ C n$. >From $x m = P C m x 1 ∈ C m ⊂ C n$ and (8), we have that
$∥ x m − P C n x 1 ∥ 2 + ∥ P C n x 1 − x 1 ∥ 2 ≤ ∥ x 1 − x m ∥ 2 .$
This implies that
$∥ x m − x n ∥ 2 ≤ ∥ x 1 − x m ∥ 2 − ∥ x n − x 1 ∥ 2 ≤ c 2 − ∥ x n − x 1 ∥ 2 .$
Since $c 2 − ∥ x n − x 1 ∥ 2 → 0$ as $n → ∞$, we have that ${ x n }$ is a Caushy sequence. Since H is complete and C is closed, there exists a point $u ∈ C$ such that $lim n → ∞ x n = u$.
Using (18), we have $lim n → ∞ ∥ x n + 1 − x n ∥ = 0$. By $x n + 1 ∈ C n + 1$, we get that
$∥ y n − x n ∥ ≤ ∥ y n − x n + 1 ∥ + ∥ x n + 1 − x n ∥ ≤ ∥ x n − x n + 1 ∥ + ∥ x n + 1 − x n ∥ ≤ 2 ∥ x n − x n + 1 ∥ .$
This implies that
$lim n → ∞ ∥ y n − x n ∥ = 0 .$
Furthermore, we have from $x n + 1 ∈ C n + 1$ that $∥ z n − x n + 1 ∥ ≤ ∥ y n − x n + 1 ∥$. We get from $∥ y n − x n + 1 ∥ → 0$ that $∥ z n − x n + 1 ∥ → 0$. From
$∥ y n − z n ∥ ≤ ∥ y n − x n + 1 ∥ + ∥ x n + 1 − z n ∥$
we have that
$lim n → ∞ ∥ y n − z n ∥ = 0 .$
Let us show $∥ z n − u n ∥ → 0$. We have from $x n + 1 ∈ C n + 1$ that
$〈 z n − x n + 1 , z n − u n 〉 ≥ ∥ z n − u n ∥ 2 .$
Since $∥ z n − x n + 1 ∥ ∥ z n − u n ∥ ≥ 〈 z n − x n + 1 , z n − u n 〉 ≥ ∥ z n − u n ∥ 2$, we have that $∥ z n − x n + 1 ∥ ≥ ∥ z n − u n ∥$. Then we get from $∥ z n − x n + 1 ∥ → 0$ that
$lim n → ∞ ∥ z n − u n ∥ = 0 .$
Since $T j$ is $k j$-demimetric for all $j ∈ { 1 , … , M }$, we get that, for $z ∈ ∩ j = 1 M F ( T j )$,
$〈 x n − z , x n − y n 〉 = 〈 x n − z , x n − ∑ j = 1 M ξ j ( ( 1 − λ n ) I + λ n T j ) x n 〉 = ∑ j = 1 M ξ j 〈 x n − z , x n − ( ( 1 − λ n ) I + λ n T j ) x n 〉 = ∑ j = 1 M ξ j λ n 〈 x n − z , x n − T j x n 〉 ≥ ∑ j = 1 M ξ j λ n 1 − k j 2 ∥ x n − T j x n ∥ 2 ≥ ∑ j = 1 M ξ j a 1 − k j 2 ∥ x n − T j x n ∥ 2 .$
We have from $lim n → ∞ ∥ y n − x n ∥ = 0$ that
$lim n → ∞ ∥ x n − T j x n ∥ = 0 , ∀ j ∈ { 1 , … , M } .$
Since $T j$ are demiclosed for all $j ∈ { 1 , … , M }$ and $lim n → ∞ x n = u$, we have that $u ∈ ∩ j = 1 M F ( T j )$. Let us show that $u ∈ ∩ i = 1 N ( B i + G ) − 1 0$. Since $Q η n ( I − η n B i )$ is nonexpansive for all $i ∈ { 1 , … , N }$, we get that, for $z ∈ ∩ i = 1 N ( B i + G ) − 1 0$,
$〈 y n − z , y n − z n 〉 = 〈 y n − z , y n − ∑ i = 1 N σ i Q η n ( I − η n B i ) y n 〉 = ∑ i = 1 N σ i 〈 y n − z , y n − Q η n ( I − η n B i ) y n 〉 ≥ ∑ i = 1 N σ i 1 2 ∥ y n − Q η n ( I − η n B i ) y n ∥ 2 .$
We have from $lim n → ∞ ∥ y n − z n ∥ = 0$ that
$lim n → ∞ ∥ y n − Q η n ( I − η n B i ) y n ∥ = 0 , ∀ i ∈ { 1 , … , N } .$
Since ${ η n }$ is bounded, we get that there exists a subsequence ${ η n l }$ of ${ η n }$ such that $lim l → ∞ η n l = η$ and $0 < b ≤ η ≤ 2 min { μ 1 , … , μ N }$. For such $η$, we get that, for $i ∈ { 1 , … , N }$ and a subsequence ${ y n l }$ of ${ y n }$ corresponding to the sequence ${ η n l }$,
$∥ y n l − Q η ( I − η B i ) y n l ∥ ≤ ∥ y n l − Q η n l ( I − η n l B i ) y n l ∥ + ∥ Q η n l ( I − η n l B i ) y n l − Q η n l ( I − η B i ) y n l ∥ + ∥ Q η n l ( I − η B i ) y n l − Q η ( I − η B i ) y n l ∥ ≤ ∥ y n l − Q η n l ( I − η n l B i ) y n l ∥ + ∥ ( I − η n l B i ) y n l − ( I − η B i ) y n l ∥ + ∥ Q η n l ( I − η B i ) y n l − Q η ( I − η B i ) y n l ∥ ≤ ∥ y n l − Q η n l ( I − η n l B i ) y n l ∥ + | η n l − η | ∥ B i y n l ∥ + | η n l − η | η ∥ Q η ( I − η B i ) y n l − ( I − η B i ) y n l ∥ .$
On the other hand, we get that, for a fixed $y ∈ C$ and $i ∈ { 1 , … , N }$,
$b ∥ B i y n ∥ ≤ η n ∥ B i y n ∥ = ∥ η n B i y n ∥ = ∥ y n − ( y − η n B i y ) + y − η n B i y − ( y n − η n B i y n ) ∥ ≤ ∥ y n − y ∥ + η n ∥ B i y ∥ + ∥ ( I − η n B i ) y − ( I − η n B i ) y n ∥ ≤ ∥ y n − y ∥ + 2 min { μ 1 , … , μ N } ∥ B i y ∥ + ∥ y − y n ∥ .$
Since ${ y n }$ is bounded, we have that ${ B i y n }$ is bounded for all $i ∈ { 1 , … , N }$. Thus we get that
$lim l → ∞ ∥ x n l − Q η ( I − η B i ) x n l ∥ = 0 , ∀ i ∈ { 1 , … , N } .$
Since $lim l → ∞ x n l = u$ and $Q η ( I − η B i )$ are demiclosed for all $i ∈ { 1 , … , N }$, we get $u ∈ ∩ i = 1 N ( B i + G ) − 1 0$. Let us show $u ∈ A − 1 0$. We have from (22) that
$lim n → ∞ ∥ z n − u n ∥ = 0 .$
Using $r n ≥ c$, we get
$lim n → ∞ 1 r n ∥ z n − u n ∥ = 0 .$
Therefore, we have
$lim n → ∞ ∥ A r n z n ∥ = lim n → ∞ 1 r n ∥ z n − u n ∥ = 0 .$
For $( p , p * ) ∈ A$, from the monotonicity of A, we have $〈 p − u n , p * − A r n z n 〉 ≥ 0$ for all $n ∈ N$. Since $z n → u$ and hence $u n → u$, we get $〈 p − u , p * 〉 ≥ 0$. From the maximallity of A, we have $u ∈ A − 1 0$. Therefore, we have $u ∈ Ω$.
Since $z 0 = P Ω x 1$, $u ∈ Ω$ and $x n → u$, we have from (17) that
$∥ x 1 − z 0 ∥ ≤ ∥ x 1 − u ∥ = lim n → ∞ ∥ x 1 − x n ∥ ≤ ∥ x 1 − z 0 ∥ .$
Then $u = z 0 .$ Therefore, we have $x n → u = z 0$. This completes the proof. ☐

## 4. Applications

In this section, using Theorem 2, we obtain well-known and new strong convergence theorems in Hilbert spaces. We know the following lemma proved by Marino and Xu [22]; see also [23]. For the sake of completeness, we give the proof.
Lemma 5
([22,23]).Let C be a nonempty, closed and convex subset of a Hilbert space H. Let k be a real number with$0 ≤ k < 1$and let$U : C → H$be a k-strict pseudo-contraction. If$x n ⇀ u$and$x n − U x n → 0$, then$u ∈ F ( U )$.
Proof.
Let us show that a nonexpansive mapping $T : C → H$ is demiclosed. Let ${ x n }$ be a sequence in C such that $x n ⇀ u$ and $x n − T x n → 0$. We have that
$∥ u − T u ∥ 2 = ∥ u − x n + x n − T u ∥ 2 = ∥ u − x n ∥ 2 + ∥ x n − T u ∥ 2 + 2 〈 u − x n , x n − T u 〉 = ∥ u − x n ∥ 2 + ∥ x n − T x n + T x n − T u ∥ 2 + 2 〈 u − x n , x n − u + u − T u 〉 = ∥ u − x n ∥ 2 + ∥ x n − T x n ∥ 2 + ∥ T x n − T u ∥ 2 + 2 〈 x n − T x n , T x n − T u 〉 − 2 ∥ u − x n ∥ 2 + 2 〈 u − x n , u − T u 〉 ≤ ∥ u − x n ∥ 2 + ∥ x n − T x n ∥ 2 + ∥ x n − u ∥ 2 + 2 〈 x n − T x n , T x n − T u 〉 − 2 ∥ u − x n ∥ 2 + 2 〈 u − x n , u − T u 〉 = ∥ x n − T x n ∥ 2 + 2 〈 x n − T x n , T x n − T u 〉 + 2 〈 u − x n , u − T u 〉 → 0 .$
Then, $u = T u$. It is obvious that a mapping $B = I − U : C → H$ is $1 − k 2$-inverse strongly monotone. Put $α = 1 − k 2$. We have that
$α ∥ B x − B y ∥ 2 ≤ 〈 x − y , B x − B y 〉 , ∀ x , y ∈ C .$
From $U = I − B$ and (9), we have that
$I − 2 α B = I − 2 α ( I − U ) = ( 1 − 2 α ) I + 2 α U$
is nonexpansive. If $x n ⇀ u$ and $x n − U x n → 0$, then
$x n − ( ( 1 − 2 α ) I + 2 α U ) x n = 2 α ( I − U ) x n → 0 .$
Since $( 1 − 2 α ) I + 2 α U$ is nonexpansive, we have $u ∈ F ( ( 1 − 2 α ) I + 2 α U ) = F ( U )$. This implies that U is demiclosed. ☐
Furthermore, we know the following lemma from Kocourek, Takahashi, and Yao [2]; see also [24].
Lemma 6
([2,24]).Let C be a nonempty, closed and convex subset of a Hilbert space H and let$U : C → H$be generalized hybrid. If$x n ⇀ u$and$x n − U x n → 0$, then$u ∈ F ( U )$.
We prove a strong convergence theorem for a finite family of strict pseudo-contractions in a Hilbert space.
Theorem 3.
Let C be a nonempty, closed and convex subset of a Hilbert space H. Let${ k 1 , … , k M } ⊂ [ 0 , 1 )$and let${ T j } j = 1 M$be a finite family of$k j$-strict pseudo-contractions of C into itself. Assume that$∩ j = 1 M F ( T j ) ≠ ∅$. For$x 1 ∈ C$and$C 1 = C$, let${ x n }$be a sequence defined by
$y n = ∑ j = 1 M ξ j ( ( 1 − λ n ) I + λ n T j ) x n , C n + 1 = { z ∈ C n : ∥ y n − z ∥ ≤ ∥ x n − z ∥ } , x n + 1 = P C n + 1 x 1 , ∀ n ∈ N ,$
where$a ∈ R$, ${ λ n } ⊂ ( 0 , ∞ )$and${ ξ 1 , … , ξ M } ⊂ ( 0 , 1 )$satisfy the following:
(1)
$0 < a ≤ λ n ≤ min { 1 − k 1 , … , 1 − k M } , ∀ n ∈ N$;
(2)
$∑ j = 1 M ξ j = 1$.
Then${ x n }$converges strongly to a point$z 0 ∈ ∩ j = 1 M F ( T j )$, where$z 0 = P ∩ j = 1 M F ( T j ) x 1$.
Proof.
Since $T j$ is a $k j$-strict pseudo-contraction of C into itself with $F ( T j ) ≠ ∅$, from (1), $T j$ is a $k j$-demimetric mapping. Furthermore, we have from Lemma 5 that $T j$ is demiclosed. We also have that if $B i = 0$ for all $i ∈ { 1 , … , N }$ in Theorem 2, then $B i$ is a 1-inverse strongly monotone mapping. Putting $η n = 1$ for all $n ∈ N$ in Theorem 2, we have that $z n = y n$ for all $n ∈ N$. Furthermore, putting $A = G = 0$ and $η n = r n = 1$ for all $n ∈ N$ in Theorem 2, we have that
$Q ν n = J r n = I , ∀ ν n > 0 , r n > 0 .$
Then we have that $u n = z n = y n$ for all $n ∈ N$. Thus, we get the desired result from Theorem 2. ☐
As a direct result of Theorem 3, we have Theorem 1 in Introduction. We can also prove the following strong convergence theorem for a finite family of inverse strongly monotone mappings in a Hilbert space. Let g be a proper, lower semicontinuous and convex function of a Hilbert space H into $( − ∞ , ∞ ]$. The subdifferential $∂ g$ of g is defined as follows:
$∂ g ( x ) = { z ∈ H : g ( x ) + 〈 z , y − x 〉 ≤ g ( y ) , ∀ y ∈ H }$
for all $x ∈ H$. We have from Rockafellar [25] that $∂ g$ is a maximal monotone mapping. Let D be a nonempty, closed, and convex subset of a Hilbert space H and let $i D$ be the indicator function of D, i.e.,
$i D ( x ) = 0 , x ∈ D , ∞ , x ∉ D .$
Then $i D$ is a proper, lower semicontinuous and convex function on H and then the subdifferential $∂ i D$ of $i D$ is a maximal monotone mapping. Thus we define the resolvent $J λ$ of $∂ i D$ for $λ > 0$, i.e.,
$J λ x = ( I + λ ∂ i D ) − 1 x$
for all $x ∈ H$. We get that, for $x ∈ H$ and $u ∈ D$,
$u = J λ x ⟺ x ∈ u + λ ∂ i D u ⟺ x ∈ u + λ N D u ⟺ x − u ∈ λ N D u ⟺ 1 λ 〈 x − u , v − u 〉 ≤ 0 , ∀ v ∈ D ⟺ 〈 x − u , v − u 〉 ≤ 0 , ∀ v ∈ D ⟺ u = P D x ,$
where $N D u$ is the normal cone to D at u, i.e.,
$N D u = { z ∈ H : 〈 z , v − u 〉 ≤ 0 , ∀ v ∈ D } .$
Theorem 4.
Let C be a nonempty, closed and convex subset of a Hilbert space H. Let${ μ 1 , … , μ N } ⊂ ( 0 , ∞ )$. Let${ B i } i = 1 N$be a finite family of$μ i$-inverse strongly monotone mappings of C into H. Assume that$∩ i = 1 N V I ( C , B i ) ≠ ∅$. Let$x 1 ∈ C$and$C 1 = C$. Let${ x n }$be a sequence defined by
$z n = ∑ i = 1 N σ i P C ( I − η n B i ) x n , C n + 1 = { z ∈ C n : ∥ z n − z ∥ ≤ ∥ x n − z ∥ } , x n + 1 = P C n + 1 x 1 , ∀ n ∈ N ,$
where$b ∈ R$, ${ η n } ⊂ ( 0 , ∞ )$and${ σ 1 , … , σ N } ⊂ ( 0 , 1 )$satisfy the following:
(1)
$0 < b ≤ η n ≤ 2 min { μ 1 , … , μ N } , ∀ n ∈ N$;
(2)
$∑ i = 1 N σ i = 1$.
Then${ x n }$converges strongly to$z 0 ∈ ∩ i = 1 N V I ( C , B i )$, where$z 0 = P ∩ i = 1 N V I ( C , B i ) x 1$.
Proof.
Putting $G = ∂ i C$ in Theorem 2, we get that for $η n > 0$, $J η n = P C .$ Furthermore, we have $( ∂ i C ) − 1 0 = C$ and $( B i + ∂ i C ) − 1 0 = V I ( C , B i )$. In fact, we get that, for $z ∈ C$,
$z ∈ ( B i + ∂ i C ) − 1 0 ⟺ 0 ∈ B i z + ∂ i C z ⟺ 0 ∈ B i z + N C z ⟺ − B i z ∈ N C z ⟺ 〈 − B i z , v − z 〉 ≤ 0 , ∀ v ∈ C ⟺ 〈 B i z , v − z 〉 ≥ 0 , ∀ v ∈ C ⟺ z ∈ V I ( C , B i ) .$
The identity mapping I is a $1 2$-demimetric mapping of C into H. Put $T j = I$ for all $j ∈ { 1 , … , M }$ and $λ n = 1 2$ for all $n ∈ N$ in Theorem 2. Then we get that $y n = x n$ for all $n ∈ N$. Furthermore, putting $A = 0$, we have $u n = z n$. Thus, we get the desired result from Theorem 2. ☐
We prove a strong convergence theorem for a finite family of generalized hybrid mappings and a finite family of inverse strongly monotone mappings in a Hilbert space.
Theorem 5.
Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let${ μ 1 , … , μ N } ⊂ ( 0 , ∞ )$. Let${ T j } j = 1 M$be a finite family of generalized hybrid mappings of C into itself and let${ B i } i = 1 N$be a finite family of$μ i$-inverse strongly monotone mappings of C into H. Suppose that
$∩ j = 1 M F ( T j ) ∩ ( ∩ i = 1 N V I ( C , B i ) ) ≠ ∅ .$
For$x 1 ∈ C$and$C 1 = C$, let${ x n }$be a sequence defined by
$y n = ∑ j = 1 M ξ j ( ( 1 − λ n ) I + λ n T j ) x n , z n = ∑ i = 1 N σ i P C ( I − η n B i ) y n , C n + 1 = { z ∈ C n : ∥ y n − z ∥ ≤ ∥ x n − z ∥ a n d ∥ z n − z ∥ ≤ ∥ y n − z ∥ } , x n + 1 = P C n + 1 x 1 , ∀ n ∈ N ,$
where$a , b , c ∈ R$,${ λ n } , { η n } ⊂ ( 0 , ∞ )$,${ ξ 1 , … , ξ M } , { σ 1 , … , σ N } ⊂ ( 0 , 1 )$and${ α n } , { β n } , { γ n } ⊂ ( 0 , 1 )$satisfy the following conditions:
(1)
$0 < a ≤ λ n ≤ 1 , ∀ n ∈ N$;
(2)
$0 < b ≤ η n ≤ 2 min { μ 1 , … , μ N } , ∀ n ∈ N$;
(3)
$∑ j = 1 M ξ j = 1$and$∑ i = 1 N σ i = 1$.
Then${ x n }$converges strongly to a point$z 0 ∈ ∩ j = 1 M F ( T j ) ∩ ( ∩ i = 1 N V I ( C , B i ) )$, where$z 0 = P ∩ j = 1 M F ( T j ) ∩ ( ∩ i = 1 N V I ( C , B i ) ) x 1$.
Proof.
Since $T j$ is a generalized hybrid mapping of C into itself such that $F ( T j ) ≠ ∅$, from (2), $T j$ is 0-demimetric. Furthermore, from Lemma 6, $T j$ is demiclosed. Furtheremore, put $G = ∂ i C$ as in the proof of Theorem 4. Then we have that $Q η n ( I − η n B i ) = P C ( I − η n B i )$ in Theorem 2. We also have that if $A = 0$, then $J r n = I$ and $u n = z n$. Therefore, we get the desired result from Theorem 2.
We prove a strong convergence theorem for a finite family of generalized hybrid mappings and a finite family of nonexpansive mappings in a Hilbert space.
Theorem 6.
Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let${ T j } j = 1 M$be a finite family of generalized hybrid mappings of C into itself and let${ U i } i = 1 N$be a finite family of nonexpansive mappings of C into H. Suppose that$∩ j = 1 M F ( T j ) ∩ ( ∩ i = 1 N F ( U i ) ) ≠ ∅$. For$x 1 ∈ C$and$C 1 = C$, let${ x n }$be a sequence defined by
$y n = ∑ j = 1 M ξ j ( ( 1 − λ n ) I + λ n T j ) x n , z n = ∑ i = 1 N σ i ( ( 1 − η n ) I + η n U i ) y n , C n + 1 = { z ∈ C n : ∥ y n − z ∥ ≤ ∥ x n − z ∥ a n d ∥ z n − z ∥ ≤ ∥ y n − z ∥ } , x n + 1 = P C n + 1 x 1 , ∀ n ∈ N ,$
where$a , b ∈ R$, ${ λ n } , { η n } ⊂ ( 0 , ∞ )$and${ ξ 1 , … , ξ M } , { σ 1 , … , σ N } ⊂ ( 0 , 1 )$satisfy the following conditions:
(1)
$0 < a ≤ λ n ≤ 1 , ∀ n ∈ N$;
(2)
$0 < b ≤ η n ≤ 1 , ∀ n ∈ N$;
(3)
$∑ j = 1 M ξ j = 1$and$∑ i = 1 N σ i = 1$.
Then${ x n }$converges strongly to a point$z 0 ∈ ∩ j = 1 M F ( T j ) ∩ ( ∩ i = 1 N F ( U i ) )$, where$z 0 = P ∩ j = 1 M F ( T j ) ∩ ( ∩ i = 1 N F ( U i ) ) x 1$.
Proof.
As in the proof of Theorem 5, $T j$ is 0-demimetric and demiclosed. Since $U i$ is nonexpansive, $B i = I − U i$ is a $1 2$-inverse strongly monotone mapping. Furthermore, we get that
$I − η n B i = I − η n ( I − U i ) = ( 1 − η n ) I + η n U i .$
Putting $A = G = 0$, we get the desired result from Theorem 2.
We finally prove a strong convergence theorem for resolvents of a maximal monotone mapping in a Hilbert space.
Theorem 7.
Let H be a Hilbert space. Let A be a maximal monotone mapping on H and let$J r = ( I + r A ) − 1$be the resolvents of A for$r > 0$. Suppose that$A − 1 0 ≠ ∅ .$For$x 1 ∈ C$and$C 1 = C$, let${ x n }$be a sequence defined by
$u n = J r n x n , C n + 1 = { z ∈ C n : 〈 x n − z , x n − u n 〉 ≥ ∥ x n − u n ∥ 2 } , x n + 1 = P C n + 1 x 1 , ∀ n ∈ N ,$
where$c ∈ R$and${ r n } ⊂ ( 0 , ∞ )$satisfy the following:
$0 < c ≤ r n , ∀ n ∈ N .$
Then${ x n }$converges strongly to a point$z 0 ∈ A − 1 0$, where$z 0 = P A − 1 0 x 1$.
Proof.
Put $T j = I$ and $B i = 0$ for all $j ∈ { 1 , 2 , … , M }$ and $i ∈ { 1 , 2 , … , N }$ in Theorem 2. Furthermore, put $G = 0$. Then we have that $x n = y n = z n$. Thus we get the desired result from Theorem 2.

## Funding

This research received no external funding.

## Conflicts of Interest

The author declares no conflict of interest.

## References

1. Browder, F.E.; Petryshyn, W.V. Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20, 197–228. [Google Scholar] [CrossRef] [Green Version]
2. Kocourek, P.; Takahashi, W.; Yao, J.-C. Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwan. J. Math. 2010, 14, 2497–2511. [Google Scholar] [CrossRef]
3. Kosaka, F.; Takahashi, W. Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM. J. Optim. 2008, 19, 824–835. [Google Scholar] [CrossRef]
4. Kosaka, F.; Takahashi, W. Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math. (Basel) 2008, 91, 166–177. [Google Scholar] [CrossRef]
5. Takahashi, W. Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal. 2010, 11, 79–88. [Google Scholar]
6. Igarashi, T.; Takahashi, W.; Tanaka, K. Weak convergence theorems for nonspreading mappings and equilibrium problems. In Nonlinear Analysis and Optimization; Akashi, S., Takahashi, W., Tanaka, T., Eds.; Yokohama Publishers: Yokohama, Japan, 2008; pp. 75–85. [Google Scholar]
7. Aoyama, K.; Kohsaka, F.; Takahashi, W. Three generalizations of firmly nonexpansive mappings: Their relations and continuous properties. J. Nonlinear Convex Anal. 2009, 10, 131–147. [Google Scholar]
8. Takahashi, W. Convex Analysis and Approximation of Fixed Points (Japanese); Yokohama Publishers: Yokohama, Japan, 2000. [Google Scholar]
9. Takahashi, W. The split common fixed point problem and the shrinking projection method in Banach spaces. J. Convex Anal. 2017, 24, 1015–1028. [Google Scholar]
10. Takahashi, W.; Takeuchi, Y.; Kubota, R. Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2008, 341, 276–286. [Google Scholar] [CrossRef]
11. Takahashi, W. Nonlinear Functional Analysis; Yokohama Publishers: Yokohama, Japan, 2000. [Google Scholar]
12. Takahashi, W. Introduction to Nonlinear and Convex Analysis; Yokohama Publishers: Yokohama, Japan, 2009. [Google Scholar]
13. Itoh, S.; Takahashi, W. The common fixed point theory of singlevalued mappings and multivalued mappings. Pac. J. Math. 1978, 79, 493–508. [Google Scholar] [CrossRef]
14. Alsulami, S.M.; Takahashi, W. The split common null point problem for maximal monotone mappings in Hilbert spaces and applications. J. Nonlinear Convex Anal. 2014, 15, 793–808. [Google Scholar]
15. Nadezhkina, N.; Takahashi, W. Strong convergence theorem by hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 2006, 16, 1230–1241. [Google Scholar] [CrossRef]
16. Takahashi, S.; Takahashi, W.; Toyoda, M. Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 2010, 147, 27–41. [Google Scholar] [CrossRef]
17. Plubtieng, S.; Takahashi, W. Generalized split feasibility problems and weak convergence theorems in Hilbert spaces. Linear Nonlinear Anal. 2015, 1, 139–158. [Google Scholar]
18. Takahashi, W.; Xu, H.-K.; Yao, J.-C. Iterative methods for generalized split feasibility problems in Hilbert spaces. Set-Valued Var. Anal. 2015, 23, 205–221. [Google Scholar] [CrossRef]
19. Alsulami, S.M.; Takahashi, W. A strong convergence theorem by the hybrid method for finite families of nonlinear and nonself mappings in a Hilbert space. J. Nonlinear Convex Anal. 2016, 17, 2511–2527. [Google Scholar]
20. Takahashi, W.; Wen, C.-F.; Yao, J.-C. The shrinking projection method for a finite family of demimetric mappings with variational inequalty problems in a Hilbert space. Fixed Point Theory 2018, 19, 407–419. [Google Scholar] [CrossRef]
21. Browder, F.E. Nonlinear maximal monotone operators in Banach spaces. Math. Ann. 1968, 175, 89–113. [Google Scholar] [CrossRef] [Green Version]
22. Marino, G.; Xu, H.-K. Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2007, 329, 336–346. [Google Scholar] [CrossRef] [Green Version]
23. Takahashi, W.; Wong, N.-C.; Yao, J.-C. Weak and strong mean convergence theorems for extended hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 2011, 12, 553–575. [Google Scholar]
24. Takahashi, W.; Yao, J.-C.; Kocourek, K. Weak and strong convergence theorems for generalized hybrid nonself-mappings in Hilbert spaces. J. Nonlinear Convex Anal. 2010, 11, 567–586. [Google Scholar]
25. Rockafellar, R.T. On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 1970, 33, 209–216. [Google Scholar] [CrossRef] [Green Version]

## Share and Cite

MDPI and ACS Style

Takahashi, W. A Strong Convergence Theorem under a New Shrinking Projection Method for Finite Families of Nonlinear Mappings in a Hilbert Space. Mathematics 2020, 8, 435. https://doi.org/10.3390/math8030435

AMA Style

Takahashi W. A Strong Convergence Theorem under a New Shrinking Projection Method for Finite Families of Nonlinear Mappings in a Hilbert Space. Mathematics. 2020; 8(3):435. https://doi.org/10.3390/math8030435

Chicago/Turabian Style

Takahashi, Wataru. 2020. "A Strong Convergence Theorem under a New Shrinking Projection Method for Finite Families of Nonlinear Mappings in a Hilbert Space" Mathematics 8, no. 3: 435. https://doi.org/10.3390/math8030435

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.